The stationary-phase method for exponential sums with multiplicative characters.
The classical principle of stationary phase descibes the asymptotics of integrals of the form \(\int_\alpha^\beta \exp(2\pi i t f(x))\,g(x)\,dx\) as \(t\to\infty\). In a previous paper, \textit{A. Dąbrowski} and \textit{B. Fisher} [Acta Arith. 80, 1--48 (1997; Zbl 0893.11032)], extending a remark of \textit{N. Katz} [Astérisque 161--162, 105--132 (1988; Zbl 0698.14014)], formulated a \(p\)-adic version of this method and gave applications to the estimation of exponential sums. In this paper a further development is given, corresponding to the case when \(g\) is a product of multiplicative characters. This allows the author to evaluate sums of the form \[ \sum \exp(2\pi i f(x_1,\dots,x_n)/p^m)\,\prod_{i=1}^{n}\chi_i(x_i) \] where the sum is over \(x_1,\dots,x_n\in({\mathbb Z}/p^m{\mathbb Z})^\times\), the \(\chi_i\) are multiplicative characters, and \(f\in{\mathbb Q}_p[x_1,\dots,x_n,1/(x_1\cdots x_n)]\) is a Laurent polynomial satisfying certain conditions. As an application, the author shows that his results allow one to recover Mauclaire's evaluation of the Gauss sum associated to a multiplicative character, and also to improve results of \textit{R. Evans} on the \(n\)-variable Kloosterman sum twisted by multiplicative characters [Twisted hyper-Kloosterman sums over finite rings of integers, in Number theory for the millennium. I, 429--448 (2002; Zbl 1030.11038)]. Related work has been carried out by \textit{T. Cochrane} and \textit{Z. Zheng} [Acta Arith. 95, 67--95 (2000; Zbl 0956.11018)]. The approach here differs in that the author considers approximate critical points \(\pmod p^j\) where \(j\approx m/2\) for large \(m\), while Cochrane and Zheng consider critical points \(\pmod p\).
scientific article; zbMATH DE number 1886715